D’Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov–Okounkov hook length formula. We generalize and extend results of Kostant and Han.

中文翻译：

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Dedekind eta函数和D'Arcais型多项式

D'Arcais型多项式编码Dedekind eta函数的幂系数的增长和不消失的特性。它们还包括关联的Laguerre多项式。我们证明了生长条件，并将其应用到复杂的简单李代数的表示理论和分区理论上，沿Nekrasov–Okounkov钩长度公式的方向。我们概括并扩展了Kostant和Han的结果。